On the Continuity of the Jacobian of a diffeomorphism

Question

Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$ we have a map $$J\phi:U\longrightarrow GL_n(\mathbb R), x\mapsto J\phi(x).$$ Is this map continuous?

Above $J\phi(x)$ is the Jacobian of $\phi$.

Answer 1

It doesn't have to be. There exist diffeomorphisms with discontinuous derivative maps. I think you desire the condition of $C^{1}$ on $\phi$.